= Solution
A <equilibrium>[critical point] is an <equilibrium>: $\boxed{f(a,b)=g(a,b)=0}$. Put $X=x-a$, $Y=y-b$. The <Jacobian matrix> $J$ gives the first-order linearized equation $(\dot X,\dot Y)^\mathsf T=J(X,Y)^\mathsf T$. The following <eigenvalue> tests classify that linearized system. For a continuously differentiable nonlinear system, the cases with nonzero real parts persist as the usual local node, <saddle equilibrium>[saddle] or spiral classifications. The purely imaginary case needs extra nonlinear information, as explained in (iv).
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